How are polynomials like other number systems such as whole numbers and integers?

You can add, subtract, and multiply them. These three operations obey the rules for integers. There's a polynomial division algorithm that fills formally the same role as the usual division algorithm for integers. Polynomials added to, subtracted from, or multiplied by other polynomials yield only polynomials. Likewise, integers added to, subtracted from, or multiplied by other integers yield only integers.